Characters of a $C^*$- algebra

Question

By a Character of a commutative unital $C^*$-algebra $A$ means a non zero homomorphism from $A$ to $\mathbb{C}$. Is it true that characters preserves positive elements? Or if $\phi$ is a character and $p$ is a projection, is it true that $\phi(p)=0$ or $1$?

Answer 1

Characters preserve positive elements:

This is true: let $\tau: A \to \Bbb{C}$ be a character. Then it is well-known that $\tau(a^*) = \overline{\tau(a)}$ for all $a \in A$ (theorem 2.1.9 from Murphy's text on $C^*$-algebras). Hence $\tau$ preserves positive elements since $$\tau(a^*a) = \overline{\tau(a)} \tau(a) = |\tau(a)|^2 \geq 0$$ for $a \in A$.

The image of a projection under a character is $0$ or $1$:

Also this is true. Let $\tau: A \to \Bbb{C}$ be a character and $p \in A$ a projection, i.e. $p=p^2 = p^*$. Then $$\tau(p) = \tau(p^2) = \tau(p)^2 \implies \tau(p) \in \{0,1\}$$

We did not use the assumption $p=p^*$ for this.